Remark 2.1 
Actually, the fractional sub-Laplace operator on the Heisenberg group \(\mathbf {H}_n\)  can be defined according to the group Fourier transform (see, for instance, [1]}). More precisely, for a function \(f\in H^2(\mathbf {H}_n)\) , it yields
\(\left(\pi _{\lambda }\left((-\Delta _{\mathrm {H}})^{\alpha }f\right)e_k,e_{\ell }\right)_{L^2(\mathbb {R}^n)}=\left((\mu _k|\lambda |)^{\alpha }\pi _{\lambda }(f)e_k,e_{\ell }\right)_{L^2(\mathbb {R}^n)}\) 
